Abstract

We introduce the notion of intrinsic subspaces of linear and affine pair geometries, which generalizes the one of projective subspaces of projective spaces. We prove that, when the affine pair geometry is the projective geometry of a Lie algebra introduced in [W. Bertram, K.-H. Neeb, Projective completions of Jordan pairs. I. The generalized projective geometry of a Lie algebra. J. Algebra 277 (2004), 474–519. MR2067615 (2005f:17031) Zbl 02105235], such intrinsic subspaces correspond to inner ideals in the associated Jordan pair, and we investigate the case of intrinsic subspaces defined by the Peirce-decomposition which is related to 5-gradings of the projective Lie algebra. These examples, as well as the examples of general and Lagrangian flag geometries, lead to the conjecture that geometries of intrinsic subspaces tend to be themselves linear pair geometries.